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## Topological Considerations

Schuster [15] proposed to base the choice of and on the idea that an embedding using delay coordinates is a topological mapping which preserves neighbourhood relations. This means that points on the attractor in which are near to each other should also be near in the embedding space . The distance of any two points cannot decrease but only increase when one increases the embedding dimension . But if this distance increases under a change from to then it is clear that is not sufficiently large, as Fig. 6 shows.

being too small means that the attractor is projected onto a space of lower dimensionality and this projection possibly destroys neighbourhood relations, resulting in some points appearing nearer to each other in the embedding space than they actually are. (For example, may be the nearest neighbour of in although this is not true in the proper embedding space . See, again, Fig. 6 for an illustration of this observation.) If, on the other hand, is sufficiently large then the distance of any two points of the attractor in embedding space should stay the same when one changes into . Applying this geometrical point of view, one can find the proper embedding dimension by choosing initially a small value of and then increasing it systematically. One knows that the proper value of is found when all distances between any two points and do not grow any more when increasing . Practically, one constructs the quantity
 (22)

where is the -th reconstructed vector in -dimensional embedding space, , and
 (23)

measures the increase of the distance between and its -th nearest neighbour, as increases. ( is some appropriate, fixed metric in .) According to the observations stated above should be greater than or equal to one. To get a notion what happens not only to the single point and its neighbours but to all the the next step is to calculate
 (24)

which considers all reconstructed points in the embedding space and all the neighbours'' of these10 and adds up the logarithms of the ratios of the respective distances. (The number of the increases linearly with , such that there would be a trivial linear -dependence in . This is removed by dividing by .) Clearly, for equal to the proper embedding dimension and for the right sampling time , should approach zero (within the experimental and numerical errors). Thus systematic variation of and seems to enable us to find sensible values for these quantities. In fact, numerical experiments done by Schuster [15] show that one can get reasonable results when using this method.

#### Footnotes

... these10
To save computing time it is also possible to consider not all the neighbours of each but only the nearest ones.

Next: An Algorithm based on Up: Analysis of Real-World Data Previous: Basic Remarks about the   Contents
Martin_Engel 2000-05-25